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Characterization of the Dual of an Orlicz Space

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Characterization of the Dual of an Orlicz Space ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXX (1990) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATY.CZNE XXX (1990) R oseli F e r n a n d e z (Sâo Paulo) Characterization of the dual of an Orlicz space Our objective is to characterize the dual of an Orlicz space & A{X , E, fi), with the only hypothesis that (X, I , ц) is a measure space with no atoms of infinite measure. This work originates from the reading of [12] and [13] of M. M. Rao, where we found some statements which were unclear to us. In particular the characterization in [13] seems to be incomplete, possibly for some fault in the fundamental definition. Here we present another characterization. 1. Preliminaries. If ф is a nondecreasing function from [0, co[ to [0, oo], such that 0 < i l/(t0) < oo for some t0 Ф 0, the function A defined on [0, oo[ by the equality A(u) = jo ÿ{t)dt is called a Young function. (1.1) Rem ark. Let A be a Young function. Then (i) A is nondecreasing and convex; (ii) the right derivative A' of A exists, is nondecreasing and is finite-valued on [0 , b[, where b is as in (iv) below; (iii) the function A defined on [0 , oo[ as A(u) = sup {uv — A(v): pg [0 , oo[}, is a Young function, called the complementary function of A; (iv) uv ^ A(u) + A(v) for u, ve [0, oo[,with equality holding if and only if at least one of the relations v = p(u) or и = q(v) is satisfied by u and v, where b = inf (ug [0, oo[: Л(ы) = oo} and В = inf {ue [0, oo[: A(u) = oo}. In what follows А, Я, b and b will be as in (1.1), and we will write a = sup (ne [0, oo[: A(u) = 0}, â = sup {ue [0, oo[: A(u) = 0}. Moreover, (X, Z, p) will be a fixed but arbitrary measure space with no atoms of infinite measure (i.e. if FeZ and p(E) = oo, then there exists FeZ, F a E, such that 0 < p(F) < oo). Unless otherwise stated, our functions will be from X to R, and we shall employ the conventions that 0/0 = l/oo = 0 - oo = 0 and inf 0 = oo. 70 R. Fernandez The Orlicz space A is the space of all measurable functions f such that §xA(k[f\)dn < со, for some /се]0, oo[. This is a complete space with the seminorm Ц-Ц^ defined by \\f\\A = inf {/ce] 0, oo[: J A(\f\/k)dp < 1}. x (1.2) P r o p o s it io n , (i) If E e l and £,Ee££A, then \\ÇE\\A ^ bp(E); (ii) for E e l the relation p(E) < oo implies that ÇEeJ?A, the converse holding if a = 0; moreover, if p(E) = oo and a > 0, then \\ÇE\\A = 1/a; (in) for f e ^ A one has §x A(\f\)dp ^ 1 if and оп^У if \\f\W ^ (iv) if <5e[0, oo[ and /ejz?^, then A(S)-ix{{xeX: |/(x)| > <5 ll/U H 1 ; (v) if fe <fA, then 1!/||л \xAi\f\)dn+ 1 . Proof, (i) follows immediately from A(u)/u for ue]0, oo[, and for this, observe that A(u)/u is nondecreasing. To prove (iv), use (1.1 .i) and (iii). For the remaining assertions, see [2]. ■ (1.3) P r o p o s it io n . For f in 3?A there is a sequence (s„) of simple functions in S£a such that (i) 0 ^ s„ ^ sn + u for all neN, if f^ 0; (ii) If-s„\ < |/| and |s„| ^ \f\ for all neN; (iii) (s„) converges to f; (iv) (s„) converges in p-measure to / ; (v) if /се]0, oo[ is such that A (\f\/k)eth en lim J A(\f-sn\/k)dp = 0. oo x Proof. Discarding the trivial case, let \\ДАФ0. The measurability of / guarantees the existence of a sequence (s„) of simple functions satisfying (i)—(iii). Moreover, if E e l is such that/ is bounded on E, then (s„ÇE) converges uniformly to fÇE. From (ii) it follows that each sne ^ A. To establish (v) it suffices to use (ii) and (iii) and apply Lebesgue’s Convergence Theorem. It remains to prove (iv), and we first consider the case in which a > 0. Let B= {xeZ: \f(x)\ >2a\\f\\A}. Clearly (snÇBC) converges uniformly to /£ BC. By' (1.2.iv) we have p(B) < oo and hence, by Egorov’s Theorem, (s„£B) converges to f£B in /r-measure. So (iv) holds in this case. If a = 0, let ce]0 , o o [ be such that 0 < A(c) < oo. Replacing к by \\f\\A in (v) we have lim $ A(\f-sn\/\\f\\A)dp = 0, n-+ oo x and so there is a subsequence (s„J such that f A(\f-sJI\\f\\A)iii < 2-tA(c/2k). X Characterization of the dual of an Orlicz space 71 From this we see that p({xeX\ |/(x) —s„k(x)| > c\\f\\A/2k}) < \/2k for all keN, and so (s„J converges to / in /^-measure. Replace (s„) by (snJ. ■ 2. Characterization of the dual space (J?A)*. Denote by JtA the closed subspace of f£A spanned by the simple functions. In [13] the author gives a representation for x*e(JtA)* as an integral, in the sense of Dunford and Schwartz [3], relative to a measure G defined as G(E) = x*{ÇE). Clearly this defines G(X) only if ^xEf£A. Since this does not occur when a = 0 and p(X) = oo, the measure G is not defined on I in this case, and so we cannot apply the theory of [3]. Thus in [5] we introduce the concept of integration relative to a measure defined on an ideal. To facilitate the reading, we transcribe from [5] some definitions and one proposition. For this, letsé be an algebra of subsets of X, and Ж an ideal of s4 (i.e. Ж is a ring, and for Ее Ж and F еЖ one has EnFe Ж). Moreover, let G be an extended real-valued finitely additive measure defined on Ж, and let v(G, •) be the total variation of G, which is obtained by replacing I by Ж in III. 1.4 of [3]. (2.1) De f in it io n . A function s is Ж-simple if there is a pairwise disjoint, finite sequence (Elt E2, ..., En) in Ж and there exists (cl5 c2, ..., cn)eRn such that s = ^"=iC ^£.. If v(G, (xeX : s(x) = cj) < oo forct ф 0, we shall say that s is G-integrable and for Eesf we define П \sdG = £ CiG{E n Et). E i= 1 (2.2) De f in it io n . A function / is G-integrable if there is a sequence (sj of G-integrable simple functions such that (i) (sn) converges in G-measure to /, i.e. for every <5e]0, oo[ one has lim (inf{w(G, F): РеЖ and {xeX: |/„(x)-/(x)| >0} c F}) = 0; n~> <x> (ii) lim J \sn — sm\dv(G, •) = 0. m,n~* oo x If / is G-integrable and Fes/ we define j / dG = lim j sndG. E n~* со E (2.3) P r o p o sit io n (Vitali Convergence Theorem). Let f be a function and (/„) a sequence of G-integrable functions such that (i) {fn) converges in G-measure to f ; (ii) \imviG'E)^0\E\fn\dv{G, •) = 0 uniformly in neiV; (iii) for each ee]0, oo[ there is a set Fe in Ж with v(G, FJ < oo and such that j If„\dv(G, •) < e, for all neN. FÎ Then f is G-integrable and for all Е е Ж one has SfdG= lim J fJG. E n_>0° £ Next we take up our characterization of {JtA)*. 72 R. Fernandez From here on we agree that sé = Z and Z1 = {Eel: ц(Е) < oo}; moreover, if a — 0 we set Ж — Zt, and if a > О, Ж = Z. (2.4) De f in it io n . We shall say that Ge &a (X, Z, p , Ж) if G is a real­ valued, finitely additive measure defined on Ж such that (i) G « in, i.e. if ЕеZ and p(E) = 0, then G(E) = 0; (ii) 0 ^ v(G, E) < oo for all ЕеЖ; (iii) <Xq = inf{fce]0, oo[: IA(G/k, X) ^ 1} < oo, where I-A(G/k, E) = sup j £ A ( r ~ § f ) ME,): ( £ „ £ 2 .......E J e â f t , £)j, for EeZ, and @(Zl} E) is the set of all pairwise disjoint finite sequences (E1, E2, ..., En) in Zt such that (J "=1 E} c= E. Unless otherwise stated, G will denote a real-valued, finitely additive measure defined on Ж such that G « /л. (2.5) Remarks, (i) For /се]0, oo[, the function IA(G/k, •) is a real-valued finitely additive measure defined on I . (ii) If 0 < ocq < oo, then I a (G/ olg, X) ^ 1. (iii) If a g = 0, then G(E) = 0 for all EeZl. (iv) The function a<?> is a seminorm on the vector space &A(X, Z, p, Ж), and is a norm if Ж = Zl (i.e. a = 0). (v) If Ge^A(X, Z, p, Ж), then by (1.2.Ü) and (2.4.ii), every simple function in A is Jf’-simple and G-integrable.
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